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CAIRO — Emboldened by the rising power of Islamists around the region, the Palestinian militant group Hamas demanded new Israeli concessions to its security and autonomy before it halts its rocket attacks on Israel, even as the conflict took an increasing toll on Sunday.

After five days of punishing Israeli airstrikes on the Hamas-controlled Gaza Strip and no letup in the rocket fire in return, representatives of Israel and Hamas met separately with Egyptian officials in Cairo on Sunday for indirect talks about a truce.

The talks came as an Israeli bomb struck a house in Gaza on Sunday afternoon, killing 11 people, in the deadliest single strike since the conflict between Israel and Hamas escalated on Wednesday. The strike, along with several others that killed civilians across the Gaza Strip, signaled that Israel was broadening its range of targets on the fifth day of the campaign.

By the end of the day, Gaza health officials reported that 70 Palestinians had been killed in airstrikes since Wednesday, including 20 children, and that 600 had been wounded. Three Israelis have been killed and at least 79 wounded by unrelenting rocket fire out of Gaza into southern Israel and as far north as Tel Aviv.

Hamas, badly outgunned on the battlefield, appeared to be trying to exploit its increased political clout with its ideological allies in Egypt’s new Islamist-led government. The group’s leaders, rejecting Israel’s call for an immediate end to the rocket attacks, have instead laid down sweeping demands that would put Hamas in a stronger position than when the conflict began: an end to Israel’s five-year-old embargo of the Gaza Strip, a pledge by Israel not to attack again and multinational guarantees that Israel would abide by its commitments.

Prime Minister Benjamin Netanyahu of Israel stuck to his demand that all rocket fire cease before the air campaign lets up, and Israeli tanks and troops remained lined up outside Gaza on Sunday. Tens of thousands of reserve troops had been called up. “The army is prepared to significantly expand the operation,” Mr. Netanyahu said at the start of a cabinet meeting.

Reda Fahmy, a member of Egypt’s upper house of Parliament and of the nation’s dominant Islamist party, who is following the talks, said Hamas’s position was just as unequivocal. “Hamas has one clear and specific demand: for the siege to be completely lifted from Gaza,” he said. “It’s not reasonable that every now and then Israel decides to level Gaza to the ground, and then we decide to sit down and talk about it after it is done. On the Israeli part, they want to stop the missiles from one side. How is that?”

He added: “If they stop the aircraft from shooting, Hamas will then stop its missiles. But violence couldn’t be stopped from one side.”

Hamas’s aggressive stance in the cease-fire talks is the first test of the group’s belief that the Arab Spring and the rise in Islamist influence around the region have strengthened its political hand, both against Israel and against Hamas’s Palestinian rivals, who now control the West Bank with Western backing.

It also puts intense new pressure on President Mohamed Morsi of Egypt, a former leader of the Muslim Brotherhood who was known for his fiery speeches defending Hamas and denouncing Israel. Mr. Morsi must now balance the conflicting demands of an Egyptian public that is deeply sympathetic to Hamas and the Palestinian cause against Western pleadings to help broker a peace and Egypt’s need for regional stability to help revive its moribund economy.